The pathwise $C^0$-Flux conjecture for Hamiltonian isotopies

Let $(M,\omega)$ be a closed connected symplectic manifold. Denote by $PSymp(M)$ and $PHam(M)$ the spaces of symplectic and Hamiltonian isotopies of $M$, respectively, both based at the identity, and equip them with the $C^0$ topology induced by

$$d_{C^0}^{path}(\phi^t,\psi^t)=\max_{0\leq t\leq 1}d_{C^0}(\phi^t,\psi^t).$$

Pathwise $C^0$-Flux conjecture. $PHam(M)$ is $C^0$-closed in $PSymp(M)$.

This is the path version of the $C^0$-Flux conjecture, which asks whether Hamiltonian diffeomorphisms are $C^0$-closed among symplectomorphisms isotopic to the identity. The question remains open in higher dimensions, although it is known in some special cases.